A well-posed restriction-type formulation for topology optimization

Cameron Talischi, University of Illinois at Urbana-Champaign

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This work presents a well-posed restriction-type formulation for topology optimization in which each design function is "regularized" by a compact operator as it appears in the state and constraint equations. In this manner, only designs with a certain level of regularity are permitted. A concrete and practically useful example of such an operator for topology optimization is convolution with a filter that effectively introduces a characteristic length scale in the problem. Another feature of the proposed formulation is the use of the Heaviside function that allows for a sharp definition of the design boundary. We prove existence of solutions under weak continuity requirements of the objective function and an abstract elliptic state equation. We also discuss a convergent and stable (i.e. checkerboards-free) finite element approximation scheme with a consistent approximation of the Heaviside function. The elements of proof provide mathematical justification for heuristic regularization of the Heaviside function in many level set formulations in the literature. Details of the optimization algorithm for the resulting discrete system along with a benchmark numerical example will be presented.

Abstract Author(s): Cameron Talischi, Glaucio H. Paulino